In the image, you have two parallel lines cut by a transversal, forming corresponding angles that are equal. The given angles are expressions involving variables $x$ and $y$.

1. First pair of corresponding angles:

   • One angle is labeled $(10x)^\circ$ and its corresponding angle across the transversal is $(2x + 80)^\circ$.
   • Since these angles are corresponding, they must be equal: $10x = 2x + 80$ Solving this: $10x - 2x = 80$ $8x = 80$ $x = 10$

2. Second pair of corresponding angles:

   • The second pair of angles is labeled $(y + 3)^\circ$ and an angle vertically opposite to it, also $(2x + 80)^\circ$.
   • Since vertical angles are equal: $y + 3 = 2x + 80$ Substituting $x = 10$ from the first equation: $y + 3 = 2(10) + 80$ $y + 3 = 20 + 80$ $y + 3 = 100$ $y = 97$

Solution:

• $x = 10$
• $y = 97$

The reasoning is based on the properties of corresponding and vertical angles, which state that corresponding angles are equal when two parallel lines are cut by a transversal, and vertical angles are always equal.

Would you like more details on this, or have any questions?

Here are some related questions:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. How do we prove that corresponding angles are equal in parallel lines?
3. Can we solve this problem using alternate exterior angles instead?
4. What happens to the angles if the lines are not parallel?
5. How does this problem change if we add another transversal?

Tip: In any geometry problem involving parallel lines, focus on angle pairs like corresponding, alternate interior, or vertical angles to set up equations for unknowns.